Introducing “Arithmetic Calculus” with Some Applications: New Terms, Definitions, Notations and Operators

New operators are presented to introduce “arithmetic calculus”, where 1) the operators are just obvious mathematical facts, and 2) arithmetic calculus refers to summing and subtracting operations without solving equations. The sole aim of this paper is to make a case for arithmetic calculus, which is lurking in conventional mathematics and science but has no identity of its own. The underlying thinking is: 1) to shift the focus from the whole sequence to any of its single elements;and 2) to factorise each element to building blocks and rules. One outcome of this emerging calculus is to understand the interconnectivity in a family of sequences, without which they are seen as discrete entities with no interconnectivity. Arithmetic calculus is a step closer towards deriving a “Tree of Numbers” reminiscent of the Tree of Life. Another windfall outcome is to show that the deconvolution problem is explicitly well-posed but at the same time implicitly ill-conditioned;and this challenges a misconception that this problem is ill-posed. If the thinking in this paper is not new, this paper forges it through a mathematical spin by presenting new terms, definitions, notations and operators. The return for these out of the blue new aspects is far reaching.

Interval arithmetic operations for uncertainty analysis with correlated interval variables

A new interval arithmetic method is proposed to solve interval functions with correlated intervals through which the overestimation problem existing in interval analysis could be significantly alleviated. The correlation between interval parameters is defined by the multidimensional parallelepiped model which is convenient to describe the correlative and independent interval variables in a unified framework. The original interval variables with correlation are transformed into the standard space without correlation,and then the relationship between the original variables and the standard interval variables is obtained. The expressions of four basic interval arithmetic operations, namely addition, subtraction, multiplication, and division, are given in the standard space. Finally, several numerical examples and a two-step bar are used to demonstrate the effectiveness of the proposed method.

A Comparison of Arithmetic Operations for Dynamic Process Optimization Approach

A comparison of arithmetic operations of two dynamic process optimization approaches called quasi-sequential approach and reduced Sequential Quadratic Programming(rSQP)simultaneous approach with respect to equality constrained optimization problems is presented.Through the detail comparison of arithmetic operations,it is concluded that the average iteration number within differential algebraic equations(DAEs)integration of quasi-sequential approach could be regarded as a criterion.One formula is given to calculate the threshold value of average iteration number.If the average iteration number is less than the threshold value,quasi-sequential approach takes advantage of rSQP simultaneous approach which is more suitable contrarily.Two optimal control problems are given to demonstrate the usage of threshold value.For optimal control problems whose objective is to stay near desired operating point,the iteration number is usually small.Therefore,quasi-sequential approach seems more suitable for such problems.

Arithmetic Operations of Generalized Trapezoidal Picture Fuzzy Numbers by Vertex Method

In this article, we define the arithmetic operations of generalized trapezoidal picture fuzzy numbers by vertex method which is assembled on a combination of the (α, γ, β)-cut concept and standard interval analysis. Various related properties are explored. Finally, some computations of picture fuzzy functions over generalized picture fuzzy variables are illustrated by using our proposed technique.

The Arithmetic of Natural Language:Toward a typology of numeral systems

Numeral systems in natural languages show astonishing variety,though with very strong unifying tendencies that are increasing as many indigenous numeral systems disappear through language contact and globalization.Most numeral systems make use of a base,typically 10,less commonly 20,followed by a wide range of other possibilities.Higher numerals are formed from primitive lower numerals by applying the processes of addition and multiplication,in many languages also exponentiation;sometimes,however,numerals are formed from a higher numeral,using subtraction or division.Numerous complexities and idiosyncrasies are discussed,as are numeral systems that fall outside this general characterization,such as restricted numeral systems with no internal arithmetic structure,and some New Guinea extended body-part counting systems.

Group Decision-Making Model of Renal Cancer Surgery Options Using Entropy Fuzzy Element Aczel-Alsina Weighted Aggregation Operators under the Environment of Fuzzy Multi-Sets

Since existing selection methods of surgical treatment schemes of renal cancer patients mainly depend on physicians’clinical experience and judgments,the surgical treatment options of renal cancer patients lack their scientifical and reasonable information expression and group decision-making model for renal cancer patients.Fuzzy multi-sets(FMSs)have a number of properties,which make them suitable for expressing the uncertain information of medical diagnoses and treatments in group decision-making(GDM)problems.To choose the most appropriate surgical treatment scheme for a patient with localized renal cell carcinoma(RCC)(T1 stage kidney tumor),this article needs to develop an effective GDM model based on the fuzzy multivalued evaluation information of the renal cancer patients.First,we propose a conversionmethod of transforming FMSs into entropy fuzzy sets(EFSs)based on the mean and Shannon entropy of a fuzzy sequence in FMS to reasonably simplify the information expression and operations of FMSs and define the score function of an entropy fuzzy element(EFE)for ranking EFEs.Second,we present the Aczel-Alsina t-norm and t-conorm operations of EFEs and the EFE Aczel-Alsina weighted arithmetic averaging(EFEAAWAA)and EFE Aczel-Alsina weighted geometric averaging(EFEAAWGA)operators.Third,we develop a multicriteria GDM model of renal cancer surgery options in the setting of FMSs.Finally,the proposed GDM model is applied to two clinical cases of renal cancer patients to choose the best surgical treatment scheme for a renal cancer patient in the setting of FMSs.The selected results of two clinical cases verify the efficiency and rationality of the proposed GDM model in the setting of FMSs.

NEW DEMODULATION TECHNOLOGY OF OPTIMIZING ENERGY OPERATOR AND APPLICATION TO GEAR FAULT DIAGNOSIS

Although many methods have been applied to diagnose the gear thult currently, the sensitivity of them is not very good. In order to make the diagnosis methods have more excellent integrated ability in such aspects as precision, sensitivity, reliability and compact algorithm, and so on, and enlightened by the energy operator separation algorithm （EOSA）, a new demodulation method which is optimizing energy operator separation algorithm （OEOSA） is presented. In the algorithm, the non-linear differential operator is utilized to its differential equation： Choosing the unit impulse response length of filter and fixing the weighting coefficient for inportant points. The method has been applied in diagnosing tooth broden and fatiguing crack of gear faults successfully. It provides demodulation analysis of machine signal with a new approach.

Efficient quantum arithmetic operation circuits for quantum image processing

Efficient quantum circuits for arithmetic operations are vital for quantum algorithms.A fault-tolerant circuit is required for a robust quantum computing in the presence of noise.Quantum circuits based on Clifford+T gates are easily rendered faulttolerant.Therefore,reducing the T-depth and T-Count without increasing the qubit number represents vital optimization goals for quantum circuits.In this study,we propose the fault-tolerant implementations for TR and Peres gates with optimized T-depth and T-Count.Next,we design fault-tolerant circuits for quantum arithmetic operations using the TR and Peres gates.Then,we implement cyclic and complete translations of quantum images using quantum arithmetic operations,and the scalar matrix multiplication.Comparative analysis and simulation results reveal that the proposed arithmetic and image operations are efficient.For instance,cyclic translations of a quantum image produce 50%T-depth reduction relative to the previous best-known cyclic translation.

Quantitative Green's function estimates for lattice quasi-periodic Schrodinger operators

In this paper,we establish quantitative Green’s function estimates for some higher-dimensional lattice quasi-periodic(QP)Schrodinger operators.The resonances in the estimates can be described via a pair of symmetric zeros of certain functions,and the estimates apply to the sub-exponential-type non-resonance conditions.As the application of quantitative Green’s function estimates,we prove both the arithmetic version of Anderson localization and the finite volume version of(1/2-)-Holder continuity of the integrated density of states(IDS)for such QP Schrodinger operators.This gives an affirmative answer to Bourgain’s problem in Bourgain(2000).

Operator algebras associated with multiplicative convolutions of arithmetic functions

The action of N on l^2(N) is studied in association with the multiplicative structure of N. Then the maximal ideal space of the Banach algebra generated by N is homeomorphic to the product of closed unit disks indexed by primes, which reflects the fundamental theorem of arithmetic. The C*-algebra generated by N does not contain any non-zero projection of finite rank. This assertion is equivalent to the existence of infinitely many primes. The von Neumann algebra generated by N is B(l^2(N)), the set of all bounded operators on l^2(N).Moreover, the differential operator on l^2(N,1/n(n+1)) defined by ▽f = μ * f is considered, where μ is the Mbius function. It is shown that the spectrum σ(▽) contains the closure of {ζ(s)-1: Re(s) > 1}. Interesting problems concerning are discussed.

建筑工程中基于层次分析和综合权重的施工质量分析方法

针对传统的质量管理方法往往难以全面而准确地评估施工过程中的各项因素的问题,提出了一种基于层次分析和综合权重的施工质量控制方法,并进行了研究设计,包括指标体系构建、递阶层次结构的建立、指标分析,从而对整个施工过程进行质量把控。

一种基于脑电信号反馈的学习注意力提升训练方法实现

注意力对于学习者来说非常重要,注意力与脑电信号密切相关,注意力可通过训练得以提升。文章提出一种基于移动智能终端应用APP脑电信号反馈的训练系统,训练包含舒尔特方格和算数运算两种方式,训练中可实时查看训练者脑电反馈的注意力状态信息,强化训练效果。实践结果表明,注意力和脑电信号之间存在正向相关性,训练者经过长时间训练其注意力能够得到显著提升,训练还能够提高训练者快速调整其注意力状态的能力。

乡村振兴绩效评价指标体系构建与评价方法——基于自适应语言加权聚合算子

由于决策环境的复杂性以及人类思维的模糊性,人们往往会面对的语言信息多属性决策问题。为解决这一问题,我们给出一种基于自适应语言加权聚合算子的多准则决策方法,该方法可以在决策评价过程中如何降低认知偏差等因素给属性权重的可靠性和稳定性带来的影响。为此,首先提出了在新的语言集下多准则决策的聚合算子,包括语言加权几何平均算子和语言加权算术平均算子。在此基础上,针对区间变量构成的不确定语言信息集,提出一种自适应语言加权算术平均决策方法。最后利用该方法判断乡村振兴绩效优劣的合理性和实用性。

关于强化缓冲算子的研究

运用灰色系统理论,在对缓冲算子研究的基础上,构造一类具有普遍意义的强化缓冲算子,研究了它们的一些特性及内在关系,从而有效解决了冲击扰动数据序列在建模预测过程中常常出现的定量预测结果与定性分析结论不符的问题.实例验证了该算子的有效性与实用性.

基于矩阵编码的遗传算法研究

分析遗传算法求解矩阵函数的局限性,提出一种基于矩阵编码的遗传算法。定义该算法的选择算子、交叉算子、变异算子,编写各算子的Matlab函数,通过仿真求解二矩阵变量函数。实例证明,该算法能确保矩阵染色体的结构完整性,提高遗传算法的速度和寻优精度。

强化缓冲算子性质的研究

通过对平均强化缓冲算子(ASBO),几何平均强化缓冲算子(GASBO),加权平均强化缓冲算子(WASBO)和加权几何平均强化缓冲算子(WGASBO)等的特性及各种内在关系的研究,明确了当序列前一部分增长(衰减)速度过缓而后一部分增长(衰减)速度过快时,冲击扰动系统数据序列通过强化缓冲算子作用后,强化缓冲作用序列变得比较平缓,从而有效地解决了在建模预测过程中常常出现的定量预测结果与定性分析结论不符的问题.

一类新的弱化缓冲算子的构造及其应用

利用灰色系统理论,在对缓冲算子和已有弱化缓冲算子研究的基础上,构造一类新的弱化缓冲算子,并研究其一些特性及内在关系,有效解决了冲击扰动数据序列在建模预测过程中经常出现的定量预测结果与定性分析结论不符的问题.实例分析结果表明,这类新的弱化缓冲算子能显著提高数据预测模型的预测精度.

基于NSGA-II的改进多目标遗传算法

在已有多目标优化算法(NSGA-II)研究和分析的基础上,为加快收敛速度,提高收敛精度,设计了新的初始筛选机制,改进了交叉算子的系数生成,提出了更为合理的排挤机制。通过典型应用函数的计算测试,结果表明:上述改进不仅具有较高的计算效率,而且能够得到分布更为合理的解,且能保持解的多样性分布。

Sobel边缘检测算法的变异实现图像增强

简述了在传统Sobel边缘检测算法的基础上进行算法创新。在边缘检测和图像增强算法中引入了区域运算的概念。详细介绍了Sobel边缘检测算法的具体改进过程,并针对红外图像进行了实际应用。

稳态操作下万吨级聚丙烯环管反应器模型的考核与分析

建立工业聚丙烯环管反应器模型以及由该模型确定各动力学参数在反应器内的分布对优化与控制聚丙烯的工业生产至关重要.基于作者所建立的稳态操作条件下聚丙烯环管反应器模型,以福建炼油公司聚丙烯分厂环管反应器为具体模型对象,采用四阶Runge-Kutta算法对该模型方程进行了求解,并给出了模型求解的关键输入参数.模型计算结果表明,所选择的反应器内动力学参数分布均匀,模型结果与定性分析吻合;同时,对提高进料质量流量以达到提高产量的情况进行分析,计算结果也能满足现场情况.